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Theorem ltaprg 7395
Description: Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.)
Assertion
Ref Expression
ltaprg  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )

Proof of Theorem ltaprg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ltaprlem 7394 . . 3  |-  ( C  e.  P.  ->  ( A  <P  B  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) )
213ad2ant3 989 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  <P  B  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) )
3 ltexpri 7389 . . . . 5  |-  ( ( C  +P.  A ) 
<P  ( C  +P.  B
)  ->  E. x  e.  P.  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) )
43adantl 275 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( C  +P.  A
)  <P  ( C  +P.  B ) )  ->  E. x  e.  P.  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) )
5 simpl1 969 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  ->  A  e.  P. )
6 simprl 505 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  ->  x  e.  P. )
7 ltaddpr 7373 . . . . . . 7  |-  ( ( A  e.  P.  /\  x  e.  P. )  ->  A  <P  ( A  +P.  x ) )
85, 6, 7syl2anc 408 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  ->  A  <P  ( A  +P.  x ) )
9 addassprg 7355 . . . . . . . . . . . 12  |-  ( ( C  e.  P.  /\  A  e.  P.  /\  x  e.  P. )  ->  (
( C  +P.  A
)  +P.  x )  =  ( C  +P.  ( A  +P.  x ) ) )
1093com12 1170 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  C  e.  P.  /\  x  e.  P. )  ->  (
( C  +P.  A
)  +P.  x )  =  ( C  +P.  ( A  +P.  x ) ) )
11103expa 1166 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  C  e.  P. )  /\  x  e.  P. )  ->  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  ( A  +P.  x ) ) )
1211adantrr 470 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  -> 
( ( C  +P.  A )  +P.  x )  =  ( C  +P.  ( A  +P.  x ) ) )
13 simprr 506 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  -> 
( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) )
1412, 13eqtr3d 2152 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  -> 
( C  +P.  ( A  +P.  x ) )  =  ( C  +P.  B ) )
15143adantl2 1123 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  -> 
( C  +P.  ( A  +P.  x ) )  =  ( C  +P.  B ) )
16 simpl3 971 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  ->  C  e.  P. )
17 addclpr 7313 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  x  e.  P. )  ->  ( A  +P.  x
)  e.  P. )
185, 6, 17syl2anc 408 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  -> 
( A  +P.  x
)  e.  P. )
19 simpl2 970 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  ->  B  e.  P. )
20 addcanprg 7392 . . . . . . . 8  |-  ( ( C  e.  P.  /\  ( A  +P.  x )  e.  P.  /\  B  e.  P. )  ->  (
( C  +P.  ( A  +P.  x ) )  =  ( C  +P.  B )  ->  ( A  +P.  x )  =  B ) )
2116, 18, 19, 20syl3anc 1201 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  -> 
( ( C  +P.  ( A  +P.  x ) )  =  ( C  +P.  B )  -> 
( A  +P.  x
)  =  B ) )
2215, 21mpd 13 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  -> 
( A  +P.  x
)  =  B )
238, 22breqtrd 3924 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  ->  A  <P  B )
2423adantlr 468 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( C  +P.  A )  <P  ( C  +P.  B ) )  /\  ( x  e.  P.  /\  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) ) )  ->  A  <P  B )
254, 24rexlimddv 2531 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( C  +P.  A
)  <P  ( C  +P.  B ) )  ->  A  <P  B )
2625ex 114 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( C  +P.  A
)  <P  ( C  +P.  B )  ->  A  <P  B ) )
272, 26impbid 128 1  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465   E.wrex 2394   class class class wbr 3899  (class class class)co 5742   P.cnp 7067    +P. cpp 7069    <P cltp 7071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-iplp 7244  df-iltp 7246
This theorem is referenced by:  prplnqu  7396  addextpr  7397  caucvgprlemcanl  7420  caucvgprprlemnkltj  7465  caucvgprprlemnbj  7469  caucvgprprlemmu  7471  caucvgprprlemloc  7479  caucvgprprlemexbt  7482  caucvgprprlemexb  7483  caucvgprprlemaddq  7484  caucvgprprlem1  7485  caucvgprprlem2  7486  ltsrprg  7523  gt0srpr  7524  lttrsr  7538  ltsosr  7540  ltasrg  7546  prsrlt  7563  ltpsrprg  7579  map2psrprg  7581
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